well I think I told you about the disastrous approach I made to my sister...

which ended in us not speaking for a year when she said No...slightly different to yours but sisters' relationships are pretty complex, just to generalise massively. we had been close and it is getting back to being good again, but it was ghastly there for a while.
Now, three years later I'm so grateful that she didn't donate when she knew she had reservations; and also I know that there are no undercurrents when we visit with our DDs; which I strongly suspect there would have been had we had my sister as a donor.
I'm grateful that an already big leap of faith (using DE) is less complicated than it might be for having had the good fortune to find a donor who, while open to contact, really doesn't want to be part of our family...
On the other hand, I had two cousins offer and they would have been just great; but they were too old at 39 and 40....I think it would have worked as they both live far away and would have had very little to do with the child.
I think your cousin sounds far less likely to cause family rifts; and if your DH is more keen for accepting her offer, then that's a huge factor to consider...
just my 2c...